Construct Non-Hierarchical P/NBD Model for Long Timeframe Synthetic Data

In this workbook we construct the non-hierarchical P/NBD models on the synthetic data with the longer timeframe.

1 Load and Construct Datasets

1.1 Load Long-Timeframe Synthetic Transaction Data

We now want to load the CD-NOW transaction data.

Code

customer_cohortdata_tbl <- read_rds("data/longsynth_customer_cohort_data_tbl.rds")
customer_cohortdata_tbl |> glimpse()

Rows: 5,000
Columns: 5
$ customer_id     <chr> "LFC201001_0001", "LFC201001_0002", "LFC201001_0003", …
$ cohort_qtr      <chr> "2010 Q1", "2010 Q1", "2010 Q1", "2010 Q1", "2010 Q1",…
$ cohort_ym       <chr> "2010 01", "2010 01", "2010 01", "2010 01", "2010 01",…
$ first_tnx_date  <dttm> 2010-01-01 04:21:16, 2010-01-03 11:37:28, 2010-01-03 …
$ total_tnx_count <int> 3, 2, 9, 1, 4, 5, 80, 2, 3, 4, 16, 11, 22, 1, 2, 8, 4,…

Code

customer_transactions_tbl <- read_rds("data/longsynth_transaction_data_tbl.rds")
customer_transactions_tbl |> glimpse()

Rows: 43,153
Columns: 7
$ customer_id   <fct> LFC201001_0001, LFC201001_0003, LFC201001_0002, LFC20100…
$ tnx_timestamp <dttm> 2010-01-01 04:21:16, 2010-01-03 09:17:44, 2010-01-03 11…
$ tnx_dow       <fct> Fri, Sun, Sun, Tue, Wed, Wed, Fri, Sat, Sun, Mon, Tue, T…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "01", "01", "01", "01", "01", "01", "0…
$ invoice_id    <chr> "T20100101-0001", "T20100103-0001", "T20100103-0002", "T…
$ tnx_amount    <dbl> 82.97, 174.89, 126.14, 121.05, 99.23, 238.19, 54.69, 116…

Code

customer_subset_id <- read_rds("data/longsynth_customer_subset_ids.rds")
customer_subset_id |> glimpse()

 Factor w/ 5000 levels "LFC201001_0001",..: 2 8 10 14 16 17 27 30 33 35 ...

We re-produce the visualisation of the transaction times we used in previous workbooks.

Code

plot_tbl <- customer_transactions_tbl |>
  group_nest(customer_id, .key = "cust_data") |>
  filter(map_int(cust_data, nrow) > 3) |>
  slice_sample(n = 30) |>
  unnest(cust_data)

ggplot(plot_tbl, aes(x = tnx_timestamp, y = customer_id)) +
  geom_line() +
  geom_point() +
  labs(
      x = "Date",
      y = "Customer ID",
      title = "Visualisation of Customer Transaction Times"
    ) +
  theme(axis.text.y = element_text(size = 10))

1.2 Load Derived Data

Code

obs_fitdata_tbl   <- read_rds("data/longsynth_obs_fitdata_tbl.rds")
obs_validdata_tbl <- read_rds("data/longsynth_obs_validdata_tbl.rds")

customer_fit_stats_tbl <- obs_fitdata_tbl |>
  rename(x = tnx_count)

1.3 Load Subset Data

We also want to construct our data subsets for the purposes of speeding up our valuations.

Code

customer_fit_subset_tbl <- obs_fitdata_tbl |>
  filter(customer_id %in% customer_subset_id)

customer_fit_subset_tbl |> glimpse()

Rows: 1,000
Columns: 6
$ customer_id    <fct> LFC201001_0003, LFC201001_0008, LFC201001_0010, LFC2010…
$ first_tnx_date <dttm> 2010-01-03 09:17:44, 2010-01-10 12:32:30, 2010-01-12 0…
$ last_tnx_date  <dttm> 2010-07-17 16:36:08, 2010-01-19 09:01:12, 2010-03-31 1…
$ tnx_count      <dbl> 8, 1, 3, 7, 1, 0, 1, 0, 2, 3, 19, 3, 2, 16, 1, 0, 0, 20…
$ t_x            <dbl> 27.9006351, 1.2647520, 11.1995948, 5.2948422, 0.6744781…
$ T_cal          <dbl> 625.8018, 624.7825, 624.5516, 623.8087, 623.7467, 623.5…

Code

customer_valid_subset_tbl <- obs_validdata_tbl |>
  filter(customer_id %in% customer_subset_id)

customer_valid_subset_tbl |> glimpse()

Rows: 1,000
Columns: 3
$ customer_id       <fct> LFC201001_0003, LFC201001_0008, LFC201001_0010, LFC2…
$ tnx_count         <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ tnx_last_interval <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, …

We now use these datasets to set the start and end dates for our various validation methods.

Code

dates_lst <- read_rds("data/longsynth_simulation_dates.rds")

use_fit_start_date <- dates_lst$use_fit_start_date
use_fit_end_date   <- dates_lst$use_fit_end_date

use_valid_start_date <- dates_lst$use_valid_start_date
use_valid_end_date   <- dates_lst$use_valid_end_date

We now split out the transaction data into fit and validation datasets.

Code

customer_fit_transactions_tbl <- customer_transactions_tbl |>
  filter(
    customer_id %in% customer_subset_id,
    tnx_timestamp >= use_fit_start_date,
    tnx_timestamp <= use_fit_end_date
    )
  
customer_fit_transactions_tbl |> glimpse()

Rows: 8,592
Columns: 7
$ customer_id   <fct> LFC201001_0003, LFC201001_0003, LFC201001_0008, LFC20100…
$ tnx_timestamp <dttm> 2010-01-03 09:17:44, 2010-01-08 09:36:58, 2010-01-10 12…
$ tnx_dow       <fct> Sun, Fri, Sun, Tue, Sun, Sun, Mon, Tue, Tue, Wed, Thu, F…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "01", "01", "02", "02", "02", "03", "03", "03", "0…
$ invoice_id    <chr> "T20100103-0001", "T20100108-0001", "T20100110-0001", "T…
$ tnx_amount    <dbl> 174.89, 54.69, 54.44, 454.77, 74.62, 169.00, 158.28, 205…

Code

customer_valid_transactions_tbl <- customer_transactions_tbl |>
  filter(
    customer_id %in% customer_subset_id,
    tnx_timestamp >= use_valid_start_date,
    tnx_timestamp <= use_valid_end_date
    )
  
customer_valid_transactions_tbl |> glimpse()

Rows: 576
Columns: 7
$ customer_id   <fct> LFC202101_0033, LFC202111_0013, LFC201912_0021, LFC20210…
$ tnx_timestamp <dttm> 2022-01-01 03:19:56, 2022-01-01 20:12:34, 2022-01-02 05…
$ tnx_dow       <fct> Sat, Sat, Sun, Sun, Sun, Sun, Mon, Mon, Tue, Tue, Tue, T…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "01", "01", "01", "0…
$ invoice_id    <chr> "T20220101-0002", "T20220101-0007", "T20220102-0004", "T…
$ tnx_amount    <dbl> 291.91, 87.70, 236.10, 173.47, 243.91, 256.40, 312.97, 1…

Finally, we want to extract the first transaction for each customer, so we can add this data to assess our models.

Code

customer_initial_tnx_tbl <- customer_fit_transactions_tbl |>
  slice_min(n = 1, order_by = tnx_timestamp, by = customer_id)

customer_initial_tnx_tbl |> glimpse()

Rows: 1,000
Columns: 7
$ customer_id   <fct> LFC201001_0003, LFC201001_0008, LFC201001_0010, LFC20100…
$ tnx_timestamp <dttm> 2010-01-03 09:17:44, 2010-01-10 12:32:30, 2010-01-12 03…
$ tnx_dow       <fct> Sun, Sun, Tue, Sun, Sun, Tue, Mon, Sat, Mon, Thu, Fri, S…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Feb, Feb, Feb, F…
$ tnx_week      <chr> "00", "01", "02", "02", "02", "03", "04", "04", "05", "0…
$ invoice_id    <chr> "T20100103-0001", "T20100110-0001", "T20100112-0001", "T…
$ tnx_amount    <dbl> 174.89, 54.44, 454.77, 74.62, 169.00, 205.07, 113.83, 25…

We now expand out these initial transactions so that we can append them to our simulations.

Code

sim_init_tbl <- customer_initial_tnx_tbl |>
  transmute(
    customer_id,
    draw_id       = list(1:n_sim),
    tnx_timestamp,
    tnx_amount
    ) |>
  unnest(draw_id)

sim_init_tbl |> glimpse()

Rows: 2,000,000
Columns: 4
$ customer_id   <fct> LFC201001_0003, LFC201001_0003, LFC201001_0003, LFC20100…
$ draw_id       <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1…
$ tnx_timestamp <dttm> 2010-01-03 09:17:44, 2010-01-03 09:17:44, 2010-01-03 09…
$ tnx_amount    <dbl> 174.89, 174.89, 174.89, 174.89, 174.89, 174.89, 174.89, …

Before we start on that, we set a few parameters for the workbook to organise our Stan code.

Code

stan_modeldir <- "stan_models"
stan_codedir  <-   "stan_code"

2 Fit First P/NBD Model

2.1 Compile and Fit Stan Model

We now compile this model using CmdStanR.

Code

pnbd_fixed_stanmodel <- cmdstan_model(
  "stan_code/pnbd_fixed.stan",
  include_paths =   stan_codedir,
  pedantic      =           TRUE,
  dir           =  stan_modeldir
  )

We then use this compiled model with our data to produce a fit of the data.

Code

stan_modelname <- "pnbd_long_fixed1"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed   <- stanfit_seed + 1

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 1.00,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_long_fixed1_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_long_fixed1_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  pnbd_long_fixed1_stanfit <- read_rds(stanfit_object_file)
}

pnbd_long_fixed1_stanfit$print()

  variable       mean     median    sd   mad         q5        q95 rhat
 lp__      -103998.48 -103998.00 74.02 74.13 -104121.00 -103878.00 1.00
 lambda[1]       0.24       0.21  0.15  0.13       0.06       0.54 1.00
 lambda[2]       0.26       0.24  0.09  0.09       0.13       0.42 1.00
 lambda[3]       0.16       0.13  0.13  0.11       0.02       0.41 1.00
 lambda[4]       0.14       0.09  0.17  0.10       0.00       0.46 1.00
 lambda[5]       0.44       0.41  0.21  0.20       0.16       0.84 1.01
 lambda[6]       0.13       0.12  0.07  0.06       0.05       0.27 1.00
 lambda[7]       0.32       0.32  0.04  0.04       0.26       0.38 1.00
 lambda[8]       0.26       0.21  0.21  0.16       0.04       0.69 1.00
 lambda[9]       0.09       0.08  0.06  0.05       0.02       0.20 1.00
 ess_bulk ess_tail
      646     1006
     2076      934
     2904     1113
     1518      872
     2167     1178
     2190     1136
     2938     1242
     2430     1537
     1915     1025
     2427     1176

 # showing 10 of 13813 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_long_fixed1_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed1-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

2.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_long_fixed1_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Code

pnbd_long_fixed1_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_long_fixed1_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

2.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Code

pnbd_stanfit <- pnbd_long_fixed1_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_long_fixed1_assess_data_lst <- run_model_assessment(
  model_stanfit       = pnbd_stanfit,
  insample_tbl        = customer_fit_subset_tbl,
  fit_label           = "pnbd_long_fixed1",
  fit_end_dttm        = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm    = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm      = use_valid_end_date   |> as.POSIXct(),
  precompute_rootdir  = "precompute",
  data_dir            = "data",
  summary_include_tnx = FALSE,
  sim_seed            = 3010
  )

pnbd_long_fixed1_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_long_fixed1_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_long_fixed1_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_long_fixed1_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_long_fixed1_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_long_fixed1_assess_valid_simstats_tbl.rds"

2.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_long_fixed1_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

2.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_long_fixed1_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

As for our long time frame data, overall our model is working well.

3 Fit Alternate Prior Model.

We want to try an alternate prior model with a smaller co-efficient of variation to see what impact it has on our procedures.

Code

stan_modelname <- "pnbd_long_fixed2"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed   <- stanfit_seed + 1

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 0.50,
    
    mu_mn     = 0.10,
    mu_cv     = 0.50,
    )


if(!file_exists(stanfit_object_file)) {
  pnbd_long_fixed2_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_long_fixed2_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  pnbd_long_fixed2_stanfit <- read_rds(stanfit_object_file)
}

pnbd_long_fixed2_stanfit$print()

  variable       mean     median    sd   mad         q5        q95 rhat
 lp__      -185986.63 -185987.00 69.43 71.16 -186101.00 -185869.00 1.00
 lambda[1]       0.25       0.23  0.11  0.10       0.10       0.45 1.00
 lambda[2]       0.26       0.25  0.08  0.08       0.14       0.39 1.00
 lambda[3]       0.21       0.20  0.10  0.09       0.08       0.40 1.00
 lambda[4]       0.21       0.19  0.11  0.10       0.07       0.42 1.00
 lambda[5]       0.33       0.31  0.13  0.12       0.16       0.58 1.00
 lambda[6]       0.17       0.17  0.07  0.07       0.08       0.30 1.01
 lambda[7]       0.31       0.31  0.03  0.04       0.26       0.37 1.00
 lambda[8]       0.25       0.23  0.12  0.11       0.10       0.47 1.00
 lambda[9]       0.14       0.13  0.06  0.06       0.06       0.25 1.00
 ess_bulk ess_tail
      729     1232
     4256     1116
     3087     1343
     3247     1292
     3794     1524
     4228     1385
     4504     1264
     4130     1590
     3424     1205
     3535     1323

 # showing 10 of 13813 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_long_fixed2_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed2-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

3.1 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_long_fixed2_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We want to check the \(N_{eff}\) statistics also.

Code

pnbd_long_fixed2_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_long_fixed2_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

3.2 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Code

pnbd_stanfit <- pnbd_long_fixed2_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_long_fixed2_assess_data_lst <- run_model_assessment(
  model_stanfit       = pnbd_stanfit,
  insample_tbl        = customer_fit_subset_tbl,
  fit_label           = "pnbd_long_fixed2",
  fit_end_dttm        = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm    = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm      = use_valid_end_date   |> as.POSIXct(),
  precompute_rootdir  = "precompute",
  data_dir            = "data",
  summary_include_tnx = FALSE,
  sim_seed            = 3020
  )

pnbd_long_fixed2_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_long_fixed2_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_long_fixed2_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_long_fixed2_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_long_fixed2_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_long_fixed2_assess_valid_simstats_tbl.rds"

3.2.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_long_fixed2_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

3.2.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_long_fixed2_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

4 Fit Tight-Lifetime Model

We now want to try a model where we use priors with a tighter coefficient of variation for lifetime but keep the CoV for transaction frequency.

Code

stan_modelname <- "pnbd_long_fixed3"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed   <- stanfit_seed + 1

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 0.50,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_long_fixed3_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_long_fixed3_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  pnbd_long_fixed3_stanfit <- read_rds(stanfit_object_file)
}

pnbd_long_fixed3_stanfit$print()

  variable       mean     median    sd   mad         q5        q95 rhat
 lp__      -151266.37 -151266.00 74.07 74.13 -151388.05 -151145.00 1.01
 lambda[1]       0.25       0.22  0.17  0.14       0.05       0.58 1.00
 lambda[2]       0.26       0.25  0.09  0.09       0.13       0.42 1.00
 lambda[3]       0.17       0.14  0.13  0.11       0.03       0.43 1.00
 lambda[4]       0.13       0.08  0.15  0.09       0.01       0.43 1.00
 lambda[5]       0.45       0.41  0.23  0.21       0.15       0.88 1.00
 lambda[6]       0.14       0.13  0.07  0.07       0.05       0.27 1.00
 lambda[7]       0.32       0.31  0.04  0.04       0.26       0.38 1.00
 lambda[8]       0.26       0.21  0.21  0.17       0.03       0.68 1.00
 lambda[9]       0.09       0.08  0.06  0.05       0.02       0.21 1.00
 ess_bulk ess_tail
      670      992
     2419     1272
     2755     1437
     2353     1169
     2143     1163
     2535     1003
     3062     1391
     2654     1282
     2275      982
     3330     1294

 # showing 10 of 13813 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_long_fixed3_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed3-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed3-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed3-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_fixed3-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

4.1 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_long_fixed3_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We want to check the \(N_{eff}\) statistics also.

Code

pnbd_long_fixed3_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_long_fixed3_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

4.2 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Code

pnbd_stanfit <- pnbd_long_fixed3_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_long_fixed3_assess_data_lst <- run_model_assessment(
  model_stanfit       = pnbd_stanfit,
  insample_tbl        = customer_fit_subset_tbl,
  fit_label           = "pnbd_long_fixed3",
  fit_end_dttm        = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm    = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm      = use_valid_end_date   |> as.POSIXct(),
  precompute_rootdir  = "precompute",
  data_dir            = "data",
  summary_include_tnx = FALSE,
  sim_seed            = 3030
  )

pnbd_long_fixed3_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_long_fixed3_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_long_fixed3_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_long_fixed3_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_long_fixed3_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_long_fixed3_assess_valid_simstats_tbl.rds"

4.2.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_long_fixed3_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

4.2.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_long_fixed3_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

5 Compare Model Outputs

We have looked at each of the models individually, but it is also worth looking at each of the models as a group.

We now want to combine both the fit and valid transaction sets to calculate the summary statistics for both.

Code

obs_summstats_tbl <- list(
    fit   = customer_fit_transactions_tbl,
    valid = customer_valid_transactions_tbl
    ) |>
  bind_rows(.id = "assess_type") |>
  group_by(assess_type) |>
  calculate_transaction_summary_statistics() |>
  pivot_longer(
    cols      = !assess_type,
    names_to  = "label",
    values_to = "obs_value"
    )

obs_summstats_tbl |> glimpse()

Rows: 16
Columns: 3
$ assess_type <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", "v…
$ label       <chr> "p10", "p25", "p50", "p75", "p90", "p99", "total_count", "…
$ obs_value   <dbl> 1.00000, 1.00000, 2.00000, 7.00000, 16.00000, 101.15000, 8…

Code

model_assess_transactions_tbl <- dir_ls("data", regexp = "pnbd_long_fixed.*_assess_.*index") |>
  enframe(name = NULL, value = "file_path") |>
  mutate(
    model_label = str_replace(file_path, "data/pnbd_long_(.*?)_assess_.*", "\\1"),
    assess_type = if_else(str_detect(file_path, "_assess_fit_"), "fit", "valid"),
    
    assess_data = map(
      file_path, construct_model_assessment_data,
      
      .progress = "construct_assess_data"
      )
    ) |>
  select(model_label, assess_type, assess_data) |>
  unnest(assess_data)

model_assess_transactions_tbl |> glimpse()

Rows: 35,519,828
Columns: 6
$ model_label   <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed1", "fixed…
$ assess_type   <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", …
$ customer_id   <fct> LFC201001_0003, LFC201001_0003, LFC201001_0003, LFC20100…
$ draw_id       <int> 1, 2, 2, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6,…
$ tnx_timestamp <dttm> 2010-02-16 17:39:13, 2010-01-27 01:20:09, 2010-01-30 13…
$ tnx_amount    <dbl> 34.02, 8.97, 134.72, 90.52, 32.00, 19.57, 72.18, 9.66, 2…

We now want to calculate the transaction statistics on this full dataset, for each separate draw.

Code

model_assess_tbl <- model_assess_transactions_tbl |>
  group_by(model_label, assess_type, draw_id) |>
  calculate_transaction_summary_statistics()

model_assess_tbl |> glimpse()

Rows: 12,000
Columns: 11
$ model_label <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed1", "fixed1"…
$ assess_type <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", "f…
$ draw_id     <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,…
$ p10         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
$ p25         <dbl> 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2…
$ p50         <dbl> 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 3, 4, 3, 4, 3, 4, 4…
$ p75         <dbl> 10.00, 10.00, 11.00, 9.75, 10.00, 9.00, 10.00, 10.00, 10.0…
$ p90         <dbl> 30.4, 28.0, 31.0, 28.0, 27.0, 23.1, 24.0, 29.6, 26.0, 28.3…
$ p99         <dbl> 166.68, 111.00, 113.68, 108.00, 155.96, 131.54, 111.22, 11…
$ total_count <int> 8195, 7495, 7594, 7459, 8474, 7258, 7087, 8209, 8306, 8225…
$ mean_count  <dbl> 12.925868, 11.270677, 12.053968, 11.764984, 13.387046, 11.…

We now combine all this data to create a number of different comparison plots for the various summary statistics.

Code

#! echo: TRUE

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "total_count", "Total Transactions"
  )

Code

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "mean_count", "Average Transactions per Customer"
  )

Code

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "p99", "99th Percentile Count"
  )

5.1 Write Assessment Data to Disk

We now want to save the assessment data to disk.

Code

model_assess_tbl |> write_rds("data/assess_data_pnbd_long_fixed_tbl.rds")

R Environment

Code

options(width = 120L)
sessioninfo::session_info()

─ Session info ───────────────────────────────────────────────────────────────────────────────────────────────────────
 setting  value
 version  R version 4.3.1 (2023-06-16)
 os       Ubuntu 22.04.3 LTS
 system   x86_64, linux-gnu
 ui       X11
 language (EN)
 collate  en_US.UTF-8
 ctype    en_US.UTF-8
 tz       Europe/Dublin
 date     2023-12-05
 pandoc   3.1.1 @ /usr/local/bin/ (via rmarkdown)

─ Packages ───────────────────────────────────────────────────────────────────────────────────────────────────────────
 package        * version    date (UTC) lib source
 abind            1.4-5      2016-07-21 [1] RSPM (R 4.3.0)
 arrayhelpers     1.1-0      2020-02-04 [1] RSPM (R 4.3.0)
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 base64enc        0.1-3      2015-07-28 [1] RSPM (R 4.3.0)
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 shinythemes      1.2.0      2021-01-25 [1] RSPM (R 4.3.0)
 StanHeaders      2.26.28    2023-09-07 [1] RSPM (R 4.3.0)
 stringi          1.7.12     2023-01-11 [1] RSPM (R 4.3.0)
 stringr        * 1.5.0      2022-12-02 [1] RSPM (R 4.3.0)
 svUnit           1.0.6      2021-04-19 [1] RSPM (R 4.3.0)
 tensorA          0.36.2     2020-11-19 [1] RSPM (R 4.3.0)
 threejs          0.3.3      2020-01-21 [1] RSPM (R 4.3.0)
 tibble         * 3.2.1      2023-03-20 [1] RSPM (R 4.3.0)
 tidybayes      * 3.0.6      2023-08-12 [1] RSPM (R 4.3.0)
 tidyr          * 1.3.0      2023-01-24 [1] RSPM (R 4.3.0)
 tidyselect       1.2.0      2022-10-10 [1] RSPM (R 4.3.0)
 tidyverse      * 2.0.0      2023-02-22 [1] RSPM (R 4.3.0)
 timechange       0.2.0      2023-01-11 [1] RSPM (R 4.3.0)
 tzdb             0.4.0      2023-05-12 [1] RSPM (R 4.3.0)
 utf8             1.2.4      2023-10-22 [1] RSPM (R 4.3.0)
 V8               4.4.0      2023-10-09 [1] RSPM (R 4.3.0)
 vctrs            0.6.4      2023-10-12 [1] RSPM (R 4.3.0)
 vroom            1.6.4      2023-10-02 [1] RSPM (R 4.3.0)
 withr            2.5.1      2023-09-26 [1] RSPM (R 4.3.0)
 xfun             0.40       2023-08-09 [1] RSPM (R 4.3.0)
 xtable           1.8-4      2019-04-21 [1] RSPM (R 4.3.0)
 xts              0.13.1     2023-04-16 [1] RSPM (R 4.3.0)
 yaml             2.3.7      2023-01-23 [1] RSPM (R 4.3.0)
 zoo              1.8-12     2023-04-13 [1] RSPM (R 4.3.0)

 [1] /usr/local/lib/R/site-library
 [2] /usr/local/lib/R/library

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Code

options(width = 80L)